We introduce a deterministic dynamical system that appears to generate the prime numbers in their natural order. The system starts from the initial leaders 2 and 3. Each leader alternates between multiplication and addition. Multiplication generates elements in a horizontal branch, while addition generates elements in a vertical branch. Newly generated vertical elements immediately become new leaders. No divisibility tests, primality tests, sieving procedures, or explicit definitions of primality are used during execution. Computational experiments show exact agreement with the ordered sequence of prime numbers for the first 5,000 generated leaders. In particular, the 5,000th generated leader is 48,611, which coincides with the 5,000th prime. The central open problem is to explain why every newly generated leader appears to be prime.
YeXuan Chen (Mon,) studied this question.